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For physics I’d expect that the unification of quantum field theory and general relativity needs a novel input, and probably by someone without a reputation to lose. Another patent clerk perhaps …… the interesting question is how to test such a unified theory in some scenario though.
For mathematics I’d be more positive about unification because it lacks the constraints of real world measurement. The greatest burden here is to maintain logical consistency.
The sphere has more degrees of freedom ( three axes ) to express it’s symmetry than the circle ( one axis ), and a consequent richer complexity of it’s symmetry group.
But here lies a more general point : it is in the eye of the beholder, as witnessed by the variety of possible answers to the given question. But I personally go with Feynman. If you lose the mathematics you lose the understanding. So the group structure of the eight-fold way is the minimum that you need in order to keep an intellectual handle on the particle zoo. Solving a Rubik’s Cube is harder than mere rotation of a square. Rotation of a circle invokes an uncountable number of choices compared to any approximating polygon. The mathematics can give fruitful surprises ….. so I equate greater symmetry of one object over another as the greater complexity in the underlying group that one chooses to model with. Some may prefer simplicity and that can be a valid viewpoint too.
Is mathematics something inherent to nature, or is it a purely human construct?
– My belief is that is inherent in nature, to the extent that one can discover the implicit order/rules underlying explicit order/behaviours. The key here is in the act of prediction ie. extrapolating a pattern to a data set not involved in the derivation of the rules. The best data sets are in the future. Take quantum mechanics for instance with it’s great predictive power of upon groups of like microscopic objects. The scale at which QM works is well out of our intuitive scale or range and the mathematical constructs serve to align our predictions with reality eg. Dirac’s prediction of anti-matter.Why do you think math is so effective in describing the physical world around us?
– I reckon the effectiveness is due to the independence of mathematics from any one person, it’s trans-generational continuity, and much of it has arisen from careful curiosity applied to the physical world. That is our understanding, as a species, of the physical world has winnowed much grain from the chaff as it were, to produce logical constructs of high grade utility.Do you think math exists by itself without physics?
– well some mathematics does and some doesn’t. My line of thought is that mathematicians tinker with concepts and possibilities, the physicists pick up the patterns by observation which is then better informed because of the applicable mathematics. It is a spectrum based on utility.